Outside the middle school program / plane geometry 37 exercises (100% corrected)

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ABC ABC A(d AB(d 1. Drawing perpendiculars with the compass E.2340 Consider the triangle ABC below : 1 Complete the figure above using the compass and the un-graduated line. The construction lines must appear on the figure : a Draw the line perpendicular to ( AB ) passing through the point C . b Draw the line passing through B and forming a right angle with ( AC ) . c Draw the line passing through A and perpendicular to the line passing through the points B and C . 2 What do you notice? E.2821 Using the compass and the ruler, for each side of the triangle, draw the perpendicular to that side passing through the vertex opposite that side : Construction lines must be left on the figure. 2. Drawing parallels with the compass E.6556 Using the compass and unsharpened straightedge, draw the line parallel to the line ( d ) and passing through the point A . E.6557 Consider the figure below : 1 Draw the line (Δ ) parallel to the line (Δ) passing through the point A . 2 Draw the line ( d ) parallel to the line ( d ) passing through the point B . 3 a Name the point C intersection of the straight lines ( d ) and (Δ ) . b Draw the triangle ABC . https://chingmath.fr chapExoCorrec/2340 sacados/2340 ABC chapExoCorrec/2821 sacados/2821 ABC chapExoCorrec/6556 sacados/6556 A(d chapExoCorrec/6557 sacados/6557 AB(d
(dA ABCI (d(d(d==(d (t(t (d1AB1C(D1D E.1458 Using your compass, draw the par-allel to the line ( d ) passing through the point A . We’ll leave the construction lines visible E.2101 Note: All construction lines must be drawn exclusively with a compass and an unmarked ruler and must be re-tained. Let ABC be a triangle and I the midpoint of [ BC ] . Draw the line ( d ) parallel to ( AB ) passing through point I in the figure below. 3. E.1546 1 a Describe all the information pro-vided with the first figure. b What can you say about the rela-tive position of the straight lines ( d ) and (Δ) ? Quote the theorem allowing such a statement. 2 a Describe all the information pro-vided with the second figure. b What can you say about the rela-tive position of the straight lines ( t ) and ( t ) ? Quote the theorem al-lowing such a statement. E.2713 1 a Draw the line ( d 2 ) parallel to the line ( d 1 ) passing through the point A . b Draw the line ( d 3 ) parallel to the line ( d 1 ) passing through the point B . c What can you say about the relative position of the straight lines ( d 2 ) and ( d 3 ) ? Justify your answer. 2 a Draw the line (Δ 2 ) perpendicular to the line (Δ 1 ) passing through the point C . b Draw the line (Δ 3 ) perpendicular to the line (Δ 1 ) pass-ing through the point B . c What can you say about the position of the straight lines (Δ 2 ) and (Δ 3 ) ? Justify your answer. https://chingmath.fr chapExoCorrec/1458 sacados/1458 (dA chapExoCorrec/2101 sacados/2101 ABCI chapExoCorrec/1546 sacados/1546 (d(d(d==(d (t(t chapExoCorrec/2713 sacados/2713 (d1AB1C(D1D
ABCDMN ABCDE ABC 3 a Draw the line ( D 2 ) parallel to the line ( D 1 ) passing through the point C . b Draw the line ( D 3 ) perpendicular to the line ( D 1 ) pass-ing through the point D . c What can you say about the relative position of the straight lines ( D 2 ) and ( D 3 ) ? Justify your answer. E.6566 Consider the configuration given be-low where the quadrilateral ABCD is a rectangle: 1 Plots should be made with ungraded ruler and compass : a Draw the line ( d ) parallel to the line ( CD ) through the point M . b Draw the line (Δ) perpendicular to the line ( DC ) through the point N . (construction lines must be apparent) . 2 For each of the questions below, name the theorem to justify the proposed relationship : a (Δ) ⊥ ( AB ) b ( d ) == ( AB ) E.6235 We consider the configuration given below : oÓ the quadrilateral ABCD is a rectangle and the triangle BEC is isosceles with major vertex C . 1 Name the theorem for stating that the lines ( DA ) and ( CB ) are parallel to each other. 2 Name the theorem for stating that the point C belongs to the perpendicular bisector of the segment [ BE ] . 4. Carry out a construction program E.2240 Consider the triangle ABC given below : Complete the figure with the following plot program : 1 Draw the line ( d ) passing through C and perpendicular to the line ( AB ) . 2 Name M the point of intersection of ( d ) and ( AB ) . 3 Trace ( d ) such that ( d ) == ( AC ) and M ∈ ( d ) . 4 Name N the point of intersection of the line ( BC ) and ( d ) . 5 Draw the line (Δ) passing through the point B and par-allel to the line ( AC ) . https://chingmath.fr chapExoCorrec/6566 sacados/6566 ABCDMN chapExoCorrec/6235 sacados/6235 ABCDE chapExoCorrec/2240 sacados/2240 ABC
ABC ABC E.2714 Consider the triangle ABC below : Draw the following lines in the figure above : 1 Draw the line ( d ) passing through the point C and per-pendicular to the line ( AB ) . 2 Name T the point of intersection of the line ( d ) with the line ( AB ) . 3 Draw the line ( d ) passing through the point T and per-pendicular to the line ( AC ) . 4 Name M the point of intersection of the line ( d ) with the line ( AC ) . 5 Draw the line (Δ) parallel to the line ( AB ) passing through the point M . 6 Name S the intersection of the line (Δ) with the line ( BC ) . 7 Draw the line (Δ ) passing through the points S and T . E.2744 Consider the triangle ABC below : Draw the following lines in the figure above : 1 Draw the line ( d ) perpendicular to the line ( BC ) and passing through the point A . 2 Name T the point of intersection of the line ( d ) and the line ( BC ) . 3 Draw the line ( d ) parallel to the line ( AB ) and passing through the point T . 4 Name M the point of intersection of the straight lines ( d ) and ( AC ) . 5 Draw the line (Δ) passing through the point M and par-allel to the line ( BC ) . 6 Name S the point of intersection of the line ( AB ) and (Δ) . 7 Draw the line ( ST ) . E.1507 Carry out the following plot pro-gram: 1 Place three points A , B and C not aligned. 2 Draw the half-lines [ CA ) and [ CB ) . 3 Draw the segment [ AB ] . 4 Place a point I belonging to segment [ AC ] . 5 Draw the line ( d ) parallel to ( AB ) passing through the point I . 6 Draw the perpendicular to the line ( BC ) passing through the point B . E.2218 Carry out the following plot pro-gram: 1 Place three points A , B , C not aligned. 2 Draw the triangle ABC . 3 Draw the line ( d ) perpendicular to the line ( AB ) passing through the point C . 4 Name I the intersection of the straight lines ( AB ) and ( d ) . 5 Draw the line (Δ) parallel to the line ( AC ) passing through the point I . E.1505 Perform the figure corresponding to program of plot following : 1 Place three dots A , B , C non-aligned. 2 Draw the triangle ABC and draw the straight ( BC ) . 3 Draw the straight ( d ) perpendicular to the straight ( BC ) passing through the point A . 4 Name M the intersection of straight ( d ) and ( AB ) . 5 Draw the line ( d ) in parallel to the line ( BC ) passing through the point A . E.1668 1 Errors in notations and expressions punctuate the plot program below ; recopy this program correcting the er-rors : a Trace AB such that [ AB ]=7 cm . b Trace [ AX ) such that A =85 o . c Tracer [ AY ) such that B =35 o . d Call C where AX and BY intersect. e Place M center of ( AB ) . f Trace d such that d == to ( AC ) and passing through M . g Let us note L the point of intersection of ( d ) . 2 Perform the above plot program. https://chingmath.fr chapExoCorrec/2714 sacados/2714 ABC chapExoCorrec/2744 sacados/2744 ABC chapExoCorrec/1507 sacados/1507 chapExoCorrec/2218 sacados/2218 chapExoCorrec/1505 sacados/1505 chapExoCorrec/1668 sacados/1668
ABC(dI ABC(dMTS(d(AB== ABC 5. Write a construction program E.2209 Consider the figure below : 1 Two plot programs are offered. Which of the two correctly plots the figure : a Place three unaligned points and draw the triangle ABC . Place I on segment [ BC ] . Draw ( AI ) perpendicular to the line ( BC ) ; name ( d ) this line. b Place three unaligned points and draw the triangle ABC . Draw the line ( d ) passing through the point A and per-pendicular to the line ( BC ) . Name I the point of intersection of the two straight lines ( d ) and ( BC ) . 2 So what can we say: This is the point I that is used to draw the line ( AI ) , or is it the straight line ( d ) that yields I ? E.2221 Consider the triangle below where : The point M is the point of intersection of the lines (Δ) , ( d ) and ( AC ) ; The point T is the point of concurrence of the straight lines ( d ) , ( d ) and ( AB ) ; The point S is the point of concurrence of the lines (Δ) , (Δ ) and ( BC ) ; As the questions progress, the figure at the end of the exer-cise will be completed ; the aim of this exercise is to find the plotting program for reconstructing this figure. 1 First, we’ll study the different characteristics of these four straight lines ; complete the table below : This straight line passe by points Perpendicular ou parallel to straight line ( d ) ( d ) (Δ) (Δ ) Now let’s identify the order of these lines and reproduce the figure : 2 a Explain that the straight line ( d ) cannot be drawn first in the figure below. b Explain that only the line ( d ) can be drawn first. c Name the new point that appears on the figure. 3 Looking again at the table in question 1 , what is the line that can currently be drawn on the figure. 4 Draw the last two straight lines. https://chingmath.fr chapExoCorrec/2209 sacados/2209 ABC(dI chapExoCorrec/2221 sacados/2221 ABC(dMTS(d(AB== ABC
ABCMN(d ABCMH(d(BC==(d (dABCM ABC(DPN(dMCACB ABCML(d(d(BC== E.2214 We consider the following configura-tion : 1 Choose from the following three plot programs the one that will produce the following figure : a Draw the triangle ABC . Place a point N on segment [ AC ] and a point M be-longing to segment [ BC ] . Draw the line (Δ) through the points M and N per-pendicular to the line ( AC ) . Draw the line ( d ) through the points M and A perpen-dicular to the line ( BC ) . b Draw the triangle ABC . Place a point M belonging to segment [ BC ] . Draw the line ( d ) through the points A and M which is perpendicular to the line ( BC ) . Draw the line (Δ) perpendicular to the line ( AC ) through the point M . Name N the point of intersection of the lines (Δ) and ( d ) . c Draw the triangle ABC . Draw the line ( d ) perpendicular to the line ( BC ) and through the point A . Name M the point of intersection of the lines ( d ) and ( BC ) . Draw the line (Δ) perpendicular to the line ( AC ) through the point M . Name N the point of intersection of the lines (Δ) and ( AC ) . 2 Perform the chosen plot program to verify that the same figure is obtained. E.1504 Give the drawing program for the figure below : E.10783 Consider the configuration below : Write a drawing program to obtain this figure. Hint: We will begin this drawing program by ˇ placing the three points A , B , C not aligned ı. E.3781 1 Determine the drawing program of the figure below start-ing with ˇ Draw a triangle ABC isosceles rectangle in C . Place M the middle of the segment [ BC ] . ı 2 a Draw the circle C with center P and having segment [ AB ] for diameter. b What do we notice? E.1513 Give the program plotted to obtain the figure below : https://chingmath.fr chapExoCorrec/2214 sacados/2214 ABCMN(d chapExoCorrec/1504 sacados/1504 ABCMH(d(BC==(d chapExoCorrec/10783 sacados/10783 (dABCM chapExoCorrec/3781 sacados/3781 ABC(DPN(dMCACB chapExoCorrec/1513 sacados/1513 ABCML(d(d(BC==
ABCILK(d(d==(AB(d ABCMNP(d(d(AB==(d (dABCMTS(d(d==(AC==(AB ABC3cm4cmXY GHI2;5cm3cmWV ABCXI E.1514 Give the drawing program for the figure below : E.2241 Write the plot program to obtain the figure below : E.1506 Write the program to plot the fol-lowing figure : 6. Us-math E.9624 With the American notations: below is given the con-struction program of the triangle ABC : In ABC , we have : AB ⊥ BC AB = 3 cm BC = 4 cm The points X and Y are placed such that : AC ←→ XY 1 Construct the triangle DEF whose construction program is given below : DEF verifies the following conditions : DF = 4 cm ; EF = 6 cm ; DE ⊥ EF The points R and S are placed such that : ←→ RS DF 2 Using American notations, give the construction program for the GHI triangle shown opposite. E.9625 Using American notation, give the plotting program for the fig-ure opposite. 7. Around the tangent https://chingmath.fr chapExoCorrec/1514 sacados/1514 ABCILK(d(d==(AB(d chapExoCorrec/2241 sacados/2241 ABCMNP(d(d(AB==(d chapExoCorrec/1506 sacados/1506 (dABCMTS(d(d==(AC==(AB chapExoCorrec/9624 sacados/9624 ABC3cm4cmXY GHI2;5cm3cmWV chapExoCorrec/9625 sacados/9625 ABCXI
OSaut 1 Saut 2 OSaut 1 E.10818 A skier is preparing to jump over a bump and land on the other side of the slope. We assume that in mid-air, the skier’s trajectory will be a circular arc. 1 We assume that the circle that best represents the skier’s trajectory and allows him to have the best ˇ landing ı à l’has its center at O . Draw this trajectory of the skier. Note: Under ideal conditions, the skier’s trajectory will be a parabola. This curve, which is very different from a circle, will be studied in high school. Opposite is a possible parabolic trajectory of the skier. Still assuming that the skier’s trajectory is a circular arc, we want to know his trajectory during the ˇ jump 2 ı so that his landing is ˇ perfect ı but this time, we will need to determine the center of the circle of his trajectory. To do this, let’s start by analyzing the trajectory of ˇ jump 1 ı. 2 On the figure of ˇ jump 1 ı, extend the straight line direction of the skier when he takes off and note all the properties between this line, the circle, and the skier’s point of impulse. 3 Reproduce all the properties of the question 2 in order to determine the exact position of the center of this circle. Then, trace the skier’s trajectory during ˇ jump 2 ı. E.10819 In the figure below, arcs of circles have been deleted : Accurately retrace these two arcs so that the direction of the arc follows exactly that of the segment. Hint: we’ll therefore be looking for the exact position of the centers of these circular arcs. 8. Around the tangent (version 2) E.10856 Opposite, a Sangaku, a tradi-tional Japanese drawing. Reproduce this drawing in the figure below where the equilat-eral triangle and circle centers have been given. https://chingmath.fr chapExoCorrec/10818 sacados/10818 OSaut 1 Saut 2 OSaut 1 chapExoCorrec/10819 sacados/10819 chapExoCorrec/10856 sacados/10856
ABCD ABCD Indication : the precision of the contact points of the vari-ous objects in a Sangaku is paramount. 9. E.2436 Definition: let A and B be two distinct points of the plane. The single straight line passing through the middle of seg-ment [ AB ] and perpendicular to line ( AB ) is called the perpendicular bisector of segment . In the plane, we have the two segments [ AB ] and [ CD ] whose representations are given below : 1 a Using the graduated ruler and square, draw the bi-sector of each of these two segments. b Name O the point of intersection of the two bisectors. 2 a Draw the circle with center O and radius [ OA ] . b What do you notice? E.6441 In the plane, we have the two seg-ments [ AB ] and [ CD ] whose representations are given below : 1 a Using the graduated ruler and square, draw the bi-sector of each of these two segments. b Name O the point of intersection of the two bisectors. 2 a Draw the circle with center O and radius [ OA ] . b What do you notice? https://chingmath.fr chapExoCorrec/2436 sacados/2436 ABCD chapExoCorrec/6441 sacados/6441 ABCD
ABCDI DABC (dABCD E.3779 Consider the quadrilateral ABCD below ; I is the midpoint of segment [ AD ] : 1 Using the ruler and set square, draw without leaving the frame : a the perpendicular bisector of segment [ BC ] , denoted ( d 1 ) ; b the perpendicular bisector of segment [ CD ] , denoted ( d 2 ) . 2 a Label O as the intersection point of the perpendicu-lar bisectors ( d 1 ) and ( d 2 ) . b Draw the circle with center O passing through point A . What do you notice? 3 Without drawing it, justify the position of the perpendic- ular bisector of segment [ AD ] . E.5611 Consider the quadrilateral ABCD shown in the box below. 1 Using a ruler and set square, draw the perpendicular bi-sectors of each of the four sides of ABCD . 2 a What is special about these four perpendicular bi-sectors? b Draw a circle with center at the point of intersection of the perpendicular bisectors of these four sides and passing through point A . What do you notice? 10. Unclassified exercises E.1618 The figure below shows a straight line ( d ) and four points in the plane : 1 Perform the following plot program : a Draw the line ( d 1 ) perpendicular to the line ( d ) passing through the point A . b Name H the point of intersection of ( d 1 ) with the line ( d ) . c Place the point A , distinct from A , on the line ( d 1 ) verifying distance equality: AH = HA 2 Perform the following plot program : a Draw the line ( d 2 ) perpendicular to the line ( d ) passing through the point B . b Name I the point of intersection of ( d 2 ) and ( d ) . c Place the point B , distinct from B , on the straight line ( d 2 ) such that we have the following equality of distance : BI = IB 3 Carry out the same type of plot for points C and D . 4 a Draw the quadrilateral ABCD . What is its nature? b Draw the quadrilateral A B C D . What is its nature? c What can we say about these two quadrilaterals rela-tive to the line ( d ) ? https://chingmath.fr chapExoCorrec/3779 sacados/3779 ABCDI chapExoCorrec/5611 sacados/5611 DABC chapExoCorrec/1618 sacados/1618 (dABCD